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Circuit lower bounds for MerlinArthur classes
 In Proc. ACM STOC
, 2007
"... We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ..."
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We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the averagecase setting, i.e., we show that MA/1 cannot be solved on more than 1/2 + 1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudorandom generators computable using O(1) bits of advice. 1
Some results on averagecase hardness within the polynomial hierarchy
 In Proceedings of the 26th Conference on Foundations of Software Technology and Theoretical Computer Science
, 2006
"... Abstract. We prove several results about the averagecase complexity of problems in the Polynomial Hierarchy (PH). We give a connection among averagecase, worstcase, and nonuniform complexity of optimization problems. Specifically, we show that if P NP is hard in the worstcase then it is either ..."
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Abstract. We prove several results about the averagecase complexity of problems in the Polynomial Hierarchy (PH). We give a connection among averagecase, worstcase, and nonuniform complexity of optimization problems. Specifically, we show that if P NP is hard in the worstcase then it is either hard on the average (in the sense of Levin) or it is nonuniformly hard (i.e. it does not have small circuits). Recently, Gutfreund, Shaltiel and TaShma (IEEE Conference on Computational Complexity, 2005) showed an interesting worstcase to averagecase connection for languages in NP, under a notion of averagecase hardness defined using uniform adversaries. We show that extending their connection to hardness against quasipolynomial time would imply that NEXP doesn’t have polynomialsize circuits. Finally we prove an unconditional averagecase hardness result. We show that for each k, there is an explicit language in P Σ2 which is hard on average for circuits of size n k. 1
Random access to advice strings and collapsing results
"... We propose a model of computation where a Turing machine is given random access to an advice string. With random access, an advice string of exponential length becomes meaningful for polynomially bounded complexity classes. We compare the power of complexity classes under this model. It gives a more ..."
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We propose a model of computation where a Turing machine is given random access to an advice string. With random access, an advice string of exponential length becomes meaningful for polynomially bounded complexity classes. We compare the power of complexity classes under this model. It gives a more stringent notion than the usual model of computation with relativization. Under this model of random access, we prove that there exist advice strings such that the Polynomialtime Hierarchy PH and Parity Polynomialtime ⊕P all collapse to P. Our main proof technique uses the decision tree lower bounds for constant depth circuits [Yao85, Cai86, H˚as86], and the algebraic machinery of Razborov and Smolensky [Raz87, Smo87].
Stringent Relativization ∗ — A New Approach for Studying Complexity Classes —
, 2006
"... A new notion of relativization—stringent relativization—has been proposed recently [CW06] for discussing collapsing relations of complexity classes, with which we hope to open a new approach for studying complexity classes. Starting with the motivation of this notion, we discuss the meaning and impl ..."
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A new notion of relativization—stringent relativization—has been proposed recently [CW06] for discussing collapsing relations of complexity classes, with which we hope to open a new approach for studying complexity classes. Starting with the motivation of this notion, we discuss the meaning and implication of collapsing relations under the stringent relativization. 1 Introduction to Stringent Relativization The notion of relativization was introduced to demonstrate the difficulty of proving certain relations among complexity classes. For example, regarding the most famous complexity relation, that of P and NP, Baker, Gill and Solovay [BGS75] showed that we can relativize it in both ways; that is, there exist two oracles A and B such that P A = NP A (the collapsing) holds
LANGUAGES TO DIAGONALIZE AGAINST ADVICE CLASSES
"... Abstract. Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. Let poly k denote those functions in O(n k). These variants imply that DTIM ..."
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Abstract. Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. Let poly k denote those functions in O(n k). These variants imply that DTIME(nk ′ ) NE /polyk does not contain PNE, DTIME(2nk ′)/polyk does not contain EXP, SPACE(nk ′)/polyk does not contain PSPACE, uniform TC 0 /polyk does not contain CH, and uniform ACC/polyk does not contain ModPH. Consequences for selective sets are also obtained. In particular, it is shown that R DTIME(nk) T (NPsel) does not contain PNE, (Lsel) does not contain PSPACE. Finally, a circuit size hierarchy theorem is established.
Technical Reports on Mathematical and Computing Sciences: TRC168
"... For relativized arguments, we propose to restrict oracle queries to "stringent" ones; for comparing the power of two machine models relative to some oracle set, stringent relativization is to restrict machines of both types to ask queries on the same segment of the oracle. In particular, f ..."
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For relativized arguments, we propose to restrict oracle queries to "stringent" ones; for comparing the power of two machine models relative to some oracle set, stringent relativization is to restrict machines of both types to ask queries on the same segment of the oracle. In particular, for investigating polynomialtime (or polynomialsize) computability, we propose polynomial stringency, bounding query length to any fixed polynomial of input length. Under such stringent oracle access, we show an oracle G such that BPP .
Relativized Collapsing Results under Stringent Oracle Access
"... For relativized arguments, we propose to restrict oracle queries to "stringent" ones. For example, when comparing the power of two machine models relative to some oracle set X, we restrict that machines of both types ask queries from the same segment of the set X. In particular, for invest ..."
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For relativized arguments, we propose to restrict oracle queries to "stringent" ones. For example, when comparing the power of two machine models relative to some oracle set X, we restrict that machines of both types ask queries from the same segment of the set X. In particular, for investigating polynomialtime (or polynomialsize) computability, we propose polynomial stringency, bounding query length to any fixed polynomial of input length. Under such stringent oracle access, we show, for example, an oracle G such that P , for any constant d 1.
Lower Bounds on Interactive Compressibility by ConstantDepth Circuits
, 2012
"... We formulate a new connection between instance compressibility [HN10]), where the compressor uses circuits from a class C, and correlation with circuits in C. We use this connection to prove the first lower bounds on general probabilistic multiround instance compression. We show that there is no pr ..."
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We formulate a new connection between instance compressibility [HN10]), where the compressor uses circuits from a class C, and correlation with circuits in C. We use this connection to prove the first lower bounds on general probabilistic multiround instance compression. We show that there is no probabilistic multiround compression protocol for Parity in which the computationally bounded party uses a nonuniform AC 0circuit and transmits at most n/(log(n)) ω(1) bits. This result is tight, and strengthens results of Dubrov and Ishai [DI06]. We also show that a similar lower bound holds for Majority. We also consider the question of round separation, i.e., whether for each r � 1, there are functions which can be compressed better with r rounds of compression than with r − 1 rounds. We answer this question affirmatively for compression using constantdepth polynomialsize circuits. Finally, we prove the first nontrivial lower bounds for 1round compressibility of Parity by polynomial size ACC 0 [p] circuits where p is an odd prime.
Electronic Colloquium on Computational Complexity, Report No. 14 (2004) Languages to diagonalize against advice classes
"... Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. These variants imply that DTIME(nk ′ ) NE /nk does not contain PNE, DTIME(2nk ′)/nk do ..."
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Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. These variants imply that DTIME(nk ′ ) NE /nk does not contain PNE, DTIME(2nk ′)/nk does not contain EXP, SPACE(nk ′)/nk does not contain PSPACE, uniform TC 0 /nk does not contain CH, and uniform ACC/nk does not contain ModPH. Consequences for selective sets are also obtained. In particular, it is shown that R DT IME(nk) T (NPsel) does not contain